2015/12/09 by Béla Bajnok, Bajnok, Béla
Mathematics · #05B30 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG #msc:05B30
paper · pdf · doi:10.48550/arxiv.1512.02981
arxiv created 2015/12/09 · arxiv updated 2015/12/10
A Euclidean t-design, as introduced by Neumaier and Seidel (1988), is a finite set \cal X ⊂ ℝn with a weight function w: \cal X → ℝ+ for which ∑r ∈ R Wr f_Sr = ∑_\bf x ∈ \cal X w(\bf x) f(\bf x) holds for every polynomial f of total degree at most t; here R is the set of norms of the points in \cal X, Wr is the total weight of all elements of \cal X with norm r, Sr is the n-dimensional sphere of radius r centered at the origin, and f_Sr is the average of f over Sr. Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), also proved a Fisher-type inequality |\cal X| ≥ N(n,|R|,t) (assuming that the design is antipodal if t is odd). For fixed n and |R| we have N(n,|R|,t)=O(tn-1). In Part I of this paper we provide a recursive construction for Euclidean t-designs in ℝn. Namely, we show how to use certain Gauss--Jacobi quadrature formulae to "lift" a Euclidean t-design in ℝn-1 to a Euclidean t-design in ℝn, preserving both the norm spectrum R and the weight sum Wr for each r ∈ R. A Euclidean design with exactly N(n,|R|,t) points is called tight. In Part II of this paper we construct tight Euclidean designs for n=2 and every t and |R| with |R| ≤ (t+5)/(4). We also provide examples for tight Euclidean designs with (n,|R|,t) ∈ \(3,2,5),(3,3,7),(4,2,7)\.